Therefore, we can see that both move a maximum of 1 unit away from their first point, and then even out to the mentioned asymptote.
For Sn , the asymptote is .
For the terms’ calculation for given n, the asymptote is .
Therefore:
As n approaches infinity, Sn approaches 2:
,
Now, we do the same thing as before, but for with the same condition for n ():
Now we have to calculate Sn again, but for :
We plot the relation between Sn and n for this case (using Excel 2010)
Looking at the graph, we can see the same happening as in the previous graph, however Sn 's asymptote was moved 1 unit further upwards till it reached . The terms calculated for n have an asymptote again (seen from the calculations), however the 2nd term ‘jumps’ upwards before it starts decreasing.
Therefore, in our case where, as n approaches infinity, Sn approaches 3:
,
Now, we observe the same behavior with , however we choose different values of a to put instead, having in mind that n should be, again, .
Since a cannot be a negative number (limitation by the ln), we can only choose values for a which are positive, therefore we will try 5, 7, and 10.
Starting with :
Now we calculate S10 for :
We will also draw the graphs for each different value of a that we chose, so we can check if there is some pattern (using Excel 2010):
We notice that when , the asymptote of Sn (which is also approximately the value of S10) is 5.
Also, as ,
We go on with :
And now, to find S10 for :
After drawing the relation between Sn and n for (with Excel 2010) we get:
We see that as ,
We see that and the asymptote (which is also approximately the value of S10) of Sn are the same (in this case the asymptote is 7). It seems that , however we will try with our last value, and see if this is true:
For :
Now we find S10 for
Now, we draw for the last time the relation between Sn and n for (with Excel 2010 again):
Sn
We note that as ,
As we were observing all of those different values for a, we’ve noticed that as , , and also , and if we make an approximate estimation of the Sn, we could easily say that . And this is the general statement that applies in our case. However, theoretically, this only holds true when .
We will examine for different values of x, and then tell what is the true general statement for any values for x and a. For now, it is .
Now, in order to get a general statement for we will have to change the values for a and x, and then find a pattern.
We will start with , and for x we will try various positive values.
To compare right the results we’ll look correct to 6 decimal places again:
:
T8
It’s obvious that the graph keeps the same way as the one described in all previous graphs. We can see that at the 5th term the line starts following a kind of asymptote which equals 4 in this case. So it is clear that the general statement Sn = a is absolutely incorrect. However Sn seems to equal a2 and as x = 2 we can say that in this exact case Sn = ax.
We can check if this statement is true with other values of a and x. For example we can take a decimal number and keep the same value of a so we can compare with the preceding.
For
T8
As it is seen this graph differs only in that there is no crossing between the x and the T8 lines. It’s due to the fact that the decimal number we’ve taken is less than 1, and it could not be a limitation for the statement that seems to be again Sn = ax.
To be sure we can check the statement with choosing a square root for x.
For
T8
Once more the graph is moving towards a certain asymptote and the statement is again
Sn = ax. As a special case we can chose x = π.
For
T8
The graph shows no difference with the preceding ones and the statement is again Sn = ax. Now we can see if the statement and the graph are similar if we change a.
Now we start with (we look correct to 6 decimal places again)
Same values for x as before:
:
T8
Changing the value of a we can see that the two lines get further but it doesn’t change the common behavior and statement.
For
T8
This graph is similar to the case a =2 where x = 0.56 and it shows only the bigger value difference between the constant line of x and the line of summation values.
For
T8
No significant change is presented in this graph.
For
T8
All above presented cases show that the best general statement is Sn = ax, however as much as n approaches infinity there appears a certain difference between the exact values of Sn and ax, which increases with the increasing of n.
We can show 3 more cases that present different combinations of a and x values.
As we have tried decimals for x, but we haven't tried decimals for a we shall do it:
For
T8
The values of summation reach the asymptote slower than the previous cases but still it is not a limitation of the common behaviour and statement. The same is mentioned when a = π.
For
T8
For , the statement is still the same. The last check we can do is with a negative value of x, as for a we cannot do that because of the limitation if the ln function.
For
T8
Clearly it is the first graph that is distinguishable and doesn’t follow the common behavior as well as these are the first values that don’t follow the statement Sn = an. The values as well as the graph seem to oscillate under and above the 0 with increasing amplitude of oscillation as much as n approaches infinity.
As a conclusion we can say that the general statement Sn = an fulfils quite well the presented infinite summation, including the first case when we stated that Sn = a, which is a particular case when x = 1, so it is according to a1 = a.
The statement should consider the following limitations:
a ≠ 0, x ≠ 0 – as this will give us a single value “0” and the graph will be a straight line on the x axis.
a > 0, x > 0 – as a > 0 is the natural limitation of the ln, and x > 0 shows an oscillating graph.
Works Cited:
[1] "Taylor's Theorem." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011. <>.
[2] "Euler–Maclaurin formula." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011.<http://en.wikipedia.org/wiki/Euler%E2%80%93Maclaurin_formula>
[3] "Series (mathematics)." Wikipedia, the Free Encyclopedia. Web. 09 Dec. 2011. <>.